Question about units
Im new here and I wanted to know what is considered a "unit". I see people saying they bet 2 units on this game, 3 units on that game. How do you figure out what a unit is?
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I see that term also, and the only unit I'm really concerned about it the one below my belt that has given countless climaxes to the ladies.
What they're talking about has to do with money management in relation to your bankroll. Hopefully your fortunate to start off with say a 5K bankroll. You really should be wagering a percentage of your bankroll per event. I use percentages, generally 1% to 5% of my bankroll ($50 to $250 on 5K) whereas these other fellas may say 1 to 5 units, whereas the only unit I'm concerned with is ...... see above.Last edited by punchmaster; 10-13-07, 12:10 PM.Comment -
I can't speak for anyone else, but if you are talking about my post, a unit is that amount as a percentage times your bank. So, 5 units is your bankroll x 5%, 3 units is your bankroll x 3%. I use the kelly criterion and a risk of ruin % of 1%, which makes the unit size basically equal the expected value. Kelly says you should always multiply your bankroll x expected value x the risk of ruin you are willing to accept, since I use 1% as risk of ruin, the risk of ruin falls away and you are left with expected value. I am oversimplifying the formula to make is easy to explain so please any nitpickers who want to flame me for not having the formula "exactly" right please put your flame in the private zone.
In my humble opinion, your standard bet size should be 1% of your bankroll. More than that and you are overbetting your bank. If the expected value is more than slightly more than 50-50 then you can bet the expected value x your bankroll.Comment -
A more accurate way to simply state Kelly would be that a bettor should set his "to-win" amount to equal his edge.
For example, if you had an edge of 5% (what you're referring to as an expected value of 105%) you'd bet the following:- odds of +100: bet 5% of bankroll
- odds of -110: bet 5.5% of bankroll
- odds of -200: bet 10% of bankroll
- odds of +200: bet 2.5% of bankroll
You'll note that in each one of these cases a win would net you the same 5% of bankroll.
As to what you refer to as "risk-of-ruin", I'm actually not sure how this makes sense in the context of Kelly. One frequently speaks of a Kelly multiplier, such that a Kelly multiplier of 1/2 would imply a bets of (approximately) half of the full-Kelly stake, and in general a kelly multiplier of κ would imply bets of (approximately) κ × the full-Kelly stake. But of course this has nothing to do with risk-of-ruin in any traditional sense.
Actually, even from an arithmetical perspective, I'm a bit perplexed by your risk-of-ruin coefficient. You state, "Kelly says you should always multiply your bankroll x expected value x the risk of ruin you are willing to accept," and then curiously continue with "since I use 1% as risk of ruin, the risk of ruin falls away and you are left with expected value", which if I hadn't seen you write elsewhere I'd assume to be a typo.
Using Kelly or (much more typically) some fraction thereof, can frequently be a valuable risk management tool for the advantage player. However, what you've described, while perhaps (depending on how you explain your risk-of-ruin coefficient) close to Kelly for bets at odds near even, will drastically diverge as odds lengthen or shorten away from that point.Comment -
I see what the problem is. First, expected value and odds are not the same thing. Just because the book is offering +$400 does not mean that you will receive $400 for every $100 bet. Suppose Dallas vs New England has Dallas at +$400 moneyline. Just because you bet $100 on Dallas does not mean you will receive $400. Even if you could make this bet over enough number of trials to enter into the law of large numbers it does not mean you can expect $400 for every $100 bet. The odds the book gives do not reflect the true expected value. If this event occurred over enough trials to be in the law of large numbers the outcome as a percentage would not equal the odds given by the bookmaker. I know you know this. So perhaps I am using the term expected value differently than you are.
The true expected value is the winnings or losses you would have if you bet at these odds over a large number of trials. Let's say that you calculated that if this game was played 1 million times, Dallas would win the game 23% of the time. To make it simple we will use 100 games for the calculations. So, for 100 games Dallas would go 23-77. If you bet on each of these games you would end up winning $9200 on the 23 games @ +$400 and losing $7700 on the 77 games @$100. So, the profit would be $1500. Betting $100 on 100 games is $10,000 at risk, netting $1,500 means you got back $11,500. The expected value is then $11,500 for $10,000 bet or 15%. So, if your "guess" that Dallas would win this matchup 23% of the time is correct and a book is offering the matchup at +$400 then the expected value is 1.15, not 4. Of course I would NEVER bet 15% of my bank on any one bet, I don't care what Kellly says, which brings me to my second point.
Second, determining the raw kelly number is only the first step. Now we have to look at variance, and use variance to calculate risk of ruin. I am only willing to tolerate a risk of ruin near zero, so I am going to lower the kelly number quite a bit. Kelly does not care about variance, Kelly would let you take your bankroll down to 1% of its original size right before you doubled it. I would have a heart attack.
So, I want to use a bet size small enough that risk of ruin is 1%.
So, the unit sizes I give are the true expected value plugged into kelly but then using a risk of ruin of 1% to get a bet size that will keep variance very low.
This is standard practice in blackjack because the variance in blackjack is so ridiculous.Last edited by curious; 10-13-07, 03:27 PM.Comment -
While what you're describing may well be a staking strategy with which you're entirely comfortable, it is most decidedly not Kelly.
The risk-of-ruin with Kelly is zero. This is true of any percentage staking strategy.
The Kelly stake is a function of odds offered and a function of expectations (which as you've correctly pointed out is itself a function of odds offered). The correct single-bet Kelly stake as a function of edge and decimal odds is:full Kelly stake = edge / (odds - 1)
and as a function of win probability and odds is:full Kelly stake = (prob * odds - 1) / (odds - 1)
If you haven't done so already you might want to check out:
Expected Value vs Expected Growth (Kelly criterion Part I) and Maximizing Expected Growth (Kelly criterion Part II)Comment -
Please do not try to educate me on Kelly. I have been playing blackjack at the pro level for over 15 years. We live, eat, drink, breathe, and sleep kelly.
Sorry, the risk of ruin in the kelly formula is not zero, it is infinity. Kelly does not care about variance. Kelly would let you run your 1 million dollar bank to one penny right before the bank doubled to 2 million.
No one in their right mind would use Kelly without adjusting it using a risk of ruin that made sense for the variance they can tolerate.
Personally, my temperment cannot tolerate much variance, which is why I always adjust kelly so that my risk of ruin is 1%.Comment -
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Schlesinger's Risk of Ruin Formula
curious,
With respect, is it possible that you are confusing Schlesinger's Risk of Ruin Formula used by BJ players to calculate session bankroll with the Kelly Criterion used to calculate optimal staking?
matekusLast edited by matekus; 10-14-07, 01:04 PM.Comment -
No, I am not confujsing them. I am combining them. Kelly does not care about variance. Kellly would be perfectly happy for your $100,000 bankroll to go to one penny just before it doubled to $200,000. While this might be great in a computer simulation, a human being's psyche cannot handle swings like that. Using full Kelly will give you one hell of a wild ride. You have to smooth out the variance so that a human being can actually bet without having a nervous breakdown or a heart attack or jump out the window.Comment -
Maslov & Zhang (1998) "Optimal Investment Strategy for Risky Assets"
Maslov & Zhang (1998) "Optimal Investment Strategy for Risky Assets" derive a formula that explicitly includes both expectation and standard deviation.
Stake = Bankroll * (Expectation / ((Expectation ^ 2) + Variance))
Any value?
matekusComment -
The term "risk-of-ruin" refers to a probability. As such claiming that Kelly implies an "infinite risk-of-ruin" makes no more sense than claiming that a particular team has an infinite probability of winning a particular game.
Within the context of sports betting, the term "risk-of run" refers to the probability of a player losing all or substantially all of his bankroll, thus rendering him completely unable to continue his participation within the market. Now in theory, every staking strategy that specifies bet recommendations as a percentage less than 100% of a player's bankroll (e.g., Kelly, where the Kelly stake will always be specified as a percent and, except in the trivial case where win probability = 1, will only always be less than 100%) will have a zero risk-of-ruin in the strict sense (i.e., in the sense of losing a player's entire bankroll).
This can be easily show mathematically: If a player's staring bankroll is given by, B<sub>0</sub>, his stake as a fraction of bankroll is given by f (where f < 1), the his Bankroll B<sub>0</sub> after n sequential losing bets will be:B<sub>n</sub> = ( 1 - f )<sup>n</sup> * B<sub>0</sub>,
So B<sub>n</sub> → 0 only as n → ∞, which means that strict risk-of-ruin will be 0 for all finite number of trials. If the probability of losing a given single bet is p > 0 (for p = 0, the Kelly stake will necessarily be zero), then the probability of losing all n bets will be given by:Pr(lose n bets out of n) = p<sup>n</sup>
So as n → ∞, B<sub>n</sub> → 0 with probability → 0.
So in other words, the strict risk-of-ruin for any and all percentage staking strategies that only recommend stakes of < 100%, will be zero over any finite number of trials, and will approach zero as number of trials approach infinity as long as the recommended stake for a win probability of zero is zero.
Now I'll fully admit that this is only strictly true in theory. In practice, a bettor will face minimum bets sizes such that although his actual bankroll is strictly positive he still won't be able to place his desired set of bets. Furthermore, if a bettor is using the income derived from his betting as his sole means of paying rent, buying food, etc., then certainly ruinous circumstances could eventually result even though his bankroll hadn’t actually shrunk to near zero. (Although in either case it would necessarily be true that such a bettor could not have Kelly utility.)
Edward O. Thorpe, author of the famous blackjack opus Beat the Dealer, addresses this very issue:
Nevertheless, one may often choose to define risk-of-ruin in a less strict sense where, the risk-of-ruin refers to the probability of losing a given fraction of bankroll prior to making a given fraction of bankroll. Indeed for most “typical” bettors, the stake advocated by full-Kelly will frequently be “uncomfortably large”, which is why many bettors tend to bet what’s known as a “fractional Kelly”. However, it’s important to note that there’s no linear correspondence between risk-of-ruin, even loosely defined, and the Kelly multiplier. Hence, the claim that a Kelly multiplier of 1% corresponds to a risk-of-ruin of 1% is fallacious and completely unfounded. (Also note: 1. A Kelly-multiplier will result in a stake of only approximately 1% of full Kelly; and 2. the non-strict risk-of-ruin with Kelly will differ depending upon when upper and lower and bounds with which a bettor selects.)Originally posted by <a href=http://bettingtools.sbrforum.com/etc/Thorpe-Kelly-A-10.pdf TARGET=_blank>Thorpe, Ch. 3</a>
This statement is completely false. While full Kelly might not weight variance in the utility function the same way as you might, variance nevertheless does factor in. Indeed the approximation of Kelly as frequently used in blackjack and in finance (although in finance it’s known informally as “Markowitz Utility”) is given by U = EV - <sup>σ<sup>2</sup></sup>/<sub>2κ</sub> where κ is the Kelly multiplier<sup>*</sup>) makes explicit the relationship between utility and variance. In fact using what’s known as a “Taylor expansion” one can show that not only is Kelly utility a function of EV and variance, it’s also a function of skewness, kurtosis, and indeed every higher order moment. For small deviations in bankroll, one can approximate this by looking at only the first two terms of the expansion, simply EV and variance, hence the Markowitz approximation above. (One might find interesting the fact that the signs of the coefficients alternate in the utility function, meaning that as expected EV is a positive trait of an outcome set, variance is a negative, and that skewness is a positive, and kurtosis a negative. Indeed this relates to a complaint one frequently hear about the Markowitz utility curve in financial circles – it tends to produce negatively skewed and leptokurtotic or "fat-tailed" portfolios.)
As I believe I’ve shown the usage of the term risk-of-ruin is inappropriate in this context. However, even if we were to replace the term “risk-of-ruin” with “Kelly multiplier” the statement would still be fallacious. While full Kelly might not be an appropriate utility descriptor for you or I, there’s nothing inherently illogical about seeking to maximize the expected growth rate of one’s bankroll.
The problem here is that you’re simply not using the proper Kelly formula. The full Kelly stake given an edge of 15% and payout odds of +400 is not 15%, but rather 15%/4 = 3.75%. See my Kelly calculator.
With all due respect, curious, while everyone is entitled to his own opinions, everyone is not entitled to his own facts. You make the claim that, “On a game where you KNOW you have an expected value of 105% then you would bet 5% of your bankroll on that game. I agree”, but then later claim that this needs to be adjusted by multiplying through by an acceptable “risk-of-ruin”.
Well while one of these two statements may well represent your own opinion, the facts are:- full-Kelly bettor will not necessarily bet 5% of bankroll in the former case (he’ll first need to divide through by 1 - decimal odds received). Put another way, the full-Kelly is not simply equal to edge;
- the formula for fractional Kelly is slightly more complex than simply multiplying through by that fractional constant (although to be fair, it is often a decent approximation); and
- using a Kelly multiplier of κ (either correctly or simply as a linear coefficient) will not somehow magically reduce a bettor risk-of-ruin (as loosely defined above – remember that the strict risk-of-ruin is always zero for a Kelly or fractional Kelly bettor) to that fraction.
Anyway, if you want to learn more about Kelly, I suggest you read the two articles to which I linked above.
<hr>
<sup>*</sup><span style="font-size: 11px;">In a traditional Markwoitz optimization, κ would frequently be replaced by <sup>1-ρ</sup>/<sub>2ρ</sub>, where ρ is a standardized coefficient of risk-aversion, defined on the interval [0,1], where 0 corresponds to risk-neutrality and 1 corresponds to total risk aversion.Comment -
No idea why this guy's giving you so much shit Ganchrow, but I'd love to see him show how there is a risk of ruin when betting Kelly (minimum bet restrictions aside).Comment -
Maybe a practical example is in order:
Let's say you know you can consistently hit exactly 60% against the sides line in the NFL for your best pick of each week. And let's say you are laying a consistent -110 for each bet. For 800 Monte Carlo runs I show your chance of any sort of loss over the entire regular season as: 110/800 = 12.6%; losing at least 1/2 your bankroll is 15/800 = 1.9%; and losing 90% of your bankroll is 4/800 = 1%. I need more runs to really tie down the 1/2 & 1/10 loss numbers, but you get the point, it is not negligible even at a superior win statistic of 60%.
Most everybody quits in disgust when they are down 90%, and it turns out there is a very real chance (approx. one in eight) of posting some sort of loss when picking exactly at the 60% level with Kelly. BTW, your median winnings are 1.8 x original bankroll for the season.Comment -
Curious you had no shot here son.
Ganch sent the full blitz and got you 20 yards behind the line on 4th and 30.Comment -
Perhaps I'm just misinterpreting but there's really no need for a Monte Carlo on this:
17 games per season
-110 odds
60% win probability
betting fraction f of bankroll
Let W = Max # of games won so as to win no more than fraction X of bankroll
1+X ≤ (1+f*10/11)W * (1-f)(17-W)
W = (Log(X+1)-17*Log(1-f))/(Log(1+f*10/11)-Log(1-f))
We're looking for Pr(of winning of no more than fraction X of bankroll), which in Excel is given by:=BINOMDIST(W,17,0.6,1)Hence:
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Hey Ganch,
Try to solve that equation drunk. This is absurd.Comment -
Ganchrow: thanks for the math check. I indeed did have an error in my skit; neglected to post -110 on a loss, so my figures are close to yours in the single Kelly column.
I do want to point out that your +75 and +100 have cells that are similar, and I suspect an error. Likewise, your double Kelly column has many similar values where I would expect a graceful % rise rather than a stair-step. Slight errors, not nearly as bad as mine.Comment -
Your suspicion is incorrect.
Nope, no errors of which I'm aware -- slight or otherwise.
I provided all the required formulas above and I invite you to attempt to determine on your own why you assertions are invalid and the results I provided are entirely as expected. A big hint may be found in the "Remarks" of the Excel documentation on BINOMDIST.Comment -
Thanks Ganchrow,
Not sure how you make the pretty boxes in this interface, but here's the values I get with 1000 runs.
I still don't see how your +75 and +100 for 1X can be the same value. Also, I cannot understand why 2x Kelly would stair-step as in your values. I've gone over my code once again, and I cannot see any errors, and to boot most of my values agree with yours within tolerance.
BTW, the reason I Monte Carlo these things is this is a 1st run set of a more complex program to compare differing betting strategies and I would really like to run to ground if any errors really exist.
Well, that looks stupid. How you you build a matrix of cells in this #$%$ interface? Cutting and pasting didn't work. (Hint: Try the [table][/table] BBCode. -- Ganch.)Comment -
Use the [TABLE][/TABLE] BBCode.
Hint: Unless you're using a VIC-20, 1,000 Monte Carlo sims of a problem of this size should complete in a fraction of a second. If you search this forum you should be able to locate a simple Monte Carlo simulator I had written in Perl that I'm sure you could readily repurpose to your needs.
Typically, one uses a Monte Carlo simulation to render a complex system more tractable.
This, however, is not a complex system but rather one whose results can readily be determined from first principles alone.
Where you're like getting tripped up with this is that the variable W (the max # of bets needed to be won so as to appreciate one's bankroll by no more than the specified fraction) may have any real value. In keeping with the binomial distribution, the Excel BINOMDIST() function use the greatest integer lower bound of that value (see first bullet point of the "Remarks" section of the Excel documentation on BINOMDIST).
Hence, because there are only 17 possible outcomes (0-17, 1-16, 2-15 ... 16-1, 17-0, etc.) there are also only 17 possible cumulative outcome probabilities. (And if anyone reading this even considers arguing the relative sequence of wins and losses to be relevant, then he or she need to think a little harder about the commutativity of multiplication).
Perhaps the attached spreadsheet will make this easier to understand.Comment -
FWIW, after a 10,000,000 trial Monte Carlo sim I get the following results:
Comment -
"Where it comes to Kelly, it's kill or be killed."
Tell 'm =========
=========== said so
Seriously, though. Kelly is not the Holy Grail to sports betting. It begins and ends with quality wagers. So if you're good you don't really need it, and if you're no good, it will eat you alive. Either way, the only thing Kelly will do is save you some time. But that does not come free of charge. You pay for it. In stress.
Even if you appreciate the theory, you don't have to follow it blindly, like some dogma set in stone. You can play with it and create your personal application, for instance by setting values for a ceiling and a bottom. If you set your ceiling at 60% winning expectation and your bottom at 54% winning expectation, and your maximum bet size at 5% BR, then this adaptation is close enough, for me, to 1/3 Kelly:
54% - 2.0 % BR
55% - 2.5 % BR
56% - 3.0 % BR
57% - 3.5% BR
58% - 4.0% BR
59% - 4.5 % BR
60% - 5.0 % BR
Round everything down to the lower level. So 58.8% is a 4.0 % bet. And 64% is 5.0% BR. If the above is still too stressful, you can change values to 1% BR for 54% winning expectation to 4% for 60% winning expectation. Whatever fits your personality best.
Avoid Kelly unless you have a good grasp of winning expectations. The tendency is to overestimate your edge. Flat betting is safer. It's just that flat betting at 2%, in the above example, equates even a huge edge with a mere 54% winning expectation. So while flat betting is a simplification, you don't want it to become an oversimplification.Last edited by Dark Horse; 03-23-09, 03:34 AM.Comment -
As to risk of ruin. I've always felt that a winning percentage, by itself, was too one-dimensional or 'naked' for Kelly. I wanted a safety net, but didn't know what it could be. Until recently, when the topic of action points came up here. A winning percentage plus margin of victory (over the spread) is a much more reliable indicator of the strength of a system.
Action points also give a much quicker indicator of the strength of a system. Where a percentage requires a large sample size, and can't be considered reliable for a system that has gone 11-2 ATS, action points tell the real story. If that 11-2 stretch shows the wins backed by 18 actions points and the losses off by 8 actions points, that is some real meat. Now, if only I knew how to incorporate action points + winning percentage into a bet size formula.
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Dark,
I agree wholeheartedly. In my botched example which Ganchrow kindly repaired; even if you are hitting exactly 60% of your picks, you still have at least a 36% chance of having a losing season with Kelly of any version. That implies that you know a priori your win %. If you are under that number, the losses get worse. So, it really is only for the real risk takers: expect to get screwed early and often.
On the other hand, a good year can heal 3 bad ones.Comment -
Ganchrow, Thanks for your patience with me. I've found the error.
Firstly, the reason I'm only running 1000 iterations is, the money management section is a subroutine to the entire season simulator and I want to test it in situ with all the rest of the code.
Second, the error was a flag getting reset after the 1st iteration from exactly 17 games to a ratio of 1/17 games for the whole season. So, some iterations had as few as 4 games selected, others were as high as 31. As you can imagine, it tended to smooth out the results rather than the step function that should have resulted from exactly 17 games each iteration.
Thanks again for the help in debugging my buggy code.Comment
